Inverse-cube central orbit classification

For nonzero angular momentum $L$, put $u=1/r$ and $\lambda =1+2mB/L^2$. The Binet equation is $u_{\theta \theta }+\lambda u=0$. Positive $\lambda$ gives cosine scattering branches, zero gives linear reciprocal radius, and negative gives exponential combinations, with energy distinguishing sinh, exponential and cosh branches. Only intervals with $u>0$ are physical.

At $B=-L^2/(2m)$ the centrifugal and force potentials cancel. With zero radial speed, any radius is a circular solution, but there is no radial restoring force; radial-velocity perturbations are not confined.